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Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925–15932]
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e2
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Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925e15932] Asterios Pantokratoras School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
article info
abstract
Article history:
The present comment concerns some doubtful results included in the above paper. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Received 29 June 2017 Accepted 21 August 2017 Available online xxx
In the above paper the modified Rayleigh number, the Peclet number and the mixed convection parameter are defined as follows Ra ¼
gkgrc Tc mam
(1)
(2)
Tc Ra x¼± 1 þ M2 Pe
(3)
The units of temperature are ½T ¼ Kelvin ðtemperatureÞ ½m ¼ kg ðmassÞ m1 length1 sec1 time1 The units of thermal diffusivity are
where g is the gravity acceleration, k is the porous medium permeability, g is the thermal expansion coefficient, r is the fluid density, T is the fluid temperature, m is the fluid dynamic viscosity, am is the fluid thermal diffusivity and M2 is the Hartmann number. The units of gravity acceleration are ½g ¼ m ðlengthÞsec2
time2
The units of porous medium permeability are ½k ¼ m
The units of dynamic viscosity are
m Pe ¼ ∞ am
2
½r ¼ kg ðmassÞ m3 length3
length
2
The units of thermal expansion coefficient are ½g ¼ Kelvin2 temperature2 The units of density are
am ¼ m2 length2 sec1 time1 and the Hartman number is dimensionless. Taking into account the above units it is found that ½Ra ¼ Kelvin1 temperature1 m1 length1 ½Pe ¼ kg ðmassÞ m3 length3 ½x ¼ kg1 mass1 m2 length2 This means that the modified Rayleigh number, the Peclet number and the mixed convection parameter are dimensional not dimensionless as the authors claim. The following thermal boundary conditions have been used (Eq. (5a,b,c) in Ref. [1]) ðiÞ ðVWTÞ
Tðx; 0Þ Tc ¼ xm Tc
(4)
DOI of original article: http://dx.doi.org/10.1016/j.ijhydene.2017.08.116. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijhydene.2017.08.139 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Pantokratoras A, Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925e15932], International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.139
2
ðiiÞ ðVHFÞ
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e2
1 vT ðx; 0Þ ¼ xð4m1Þ=2 Tc vy
1 vT ðx; 0Þ ¼ xð2m1Þ=2 ðiiiÞ ðVHFÞ T Tc vy
(5)
(6)
c From Eq. (4) it is clear that the term Tðx;0ÞT is dimensionless. Tc m This means that the term x must be also dimensionless. The only case that the term xm is dimensionless is the case with m ¼ 0. Otherwise its units will be lengthm (the units of x are
ðx; 0Þ is length). From Eq. (5) it is found that the term 1 vT Tc dimensionless. This means that the term vyxð4m1Þ=2 must be also dimensionless. The units of x and y are length. Therefore the exponent m must be m ¼ 1=4. For the case (iii) it is found that the exponent should be m ¼ 1=2. In summary the boundary conditions 5a, 5b and 5c in Ref. [1] are valid only for the following conditions ðiÞ ðVWTÞ m ¼ 0
(7)
ðiiÞ ðVHFÞ m ¼ 1=4
(8)
ðiiiÞ ðVHFÞ m ¼ 1=2
(9)
However, the authors presented results for values of exponent m different from the above values. For example in Fig. 6 in Ref. [1] temperature profiles are presented for the case (i) for m ¼ 0:2, m ¼ 0 and m ¼ 0:2. The results for m ¼ 0:2 and m ¼ 0:2 are wrong. Taking into account all the above the results presented in Ref. [1] are doubtful.
reference
[1] Haq Rizwan Ul, Hamouch Zakia, Hussain ST, Mekkaoui Toufik. MHD mixed convection flow along a vertically heated sheet. Int J Hydrogen Energy 2017;42:15925e32.
Please cite this article in press as: Pantokratoras A, Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925e15932], International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.139
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925e15932] Asterios Pantokratoras School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
article info
abstract
Article history:
The present comment concerns some doubtful results included in the above paper. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Received 29 June 2017 Accepted 21 August 2017 Available online xxx
In the above paper the modified Rayleigh number, the Peclet number and the mixed convection parameter are defined as follows Ra ¼
gkgrc Tc mam
(1)
(2)
Tc Ra x¼± 1 þ M2 Pe
(3)
The units of temperature are ½T ¼ Kelvin ðtemperatureÞ ½m ¼ kg ðmassÞ m1 length1 sec1 time1 The units of thermal diffusivity are
where g is the gravity acceleration, k is the porous medium permeability, g is the thermal expansion coefficient, r is the fluid density, T is the fluid temperature, m is the fluid dynamic viscosity, am is the fluid thermal diffusivity and M2 is the Hartmann number. The units of gravity acceleration are ½g ¼ m ðlengthÞsec2
time2
The units of porous medium permeability are ½k ¼ m
The units of dynamic viscosity are
m Pe ¼ ∞ am
2
½r ¼ kg ðmassÞ m3 length3
length
2
The units of thermal expansion coefficient are ½g ¼ Kelvin2 temperature2 The units of density are
am ¼ m2 length2 sec1 time1 and the Hartman number is dimensionless. Taking into account the above units it is found that ½Ra ¼ Kelvin1 temperature1 m1 length1 ½Pe ¼ kg ðmassÞ m3 length3 ½x ¼ kg1 mass1 m2 length2 This means that the modified Rayleigh number, the Peclet number and the mixed convection parameter are dimensional not dimensionless as the authors claim. The following thermal boundary conditions have been used (Eq. (5a,b,c) in Ref. [1]) ðiÞ ðVWTÞ
Tðx; 0Þ Tc ¼ xm Tc
(4)
DOI of original article: http://dx.doi.org/10.1016/j.ijhydene.2017.08.116. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijhydene.2017.08.139 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Pantokratoras A, Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925e15932], International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.139
2
ðiiÞ ðVHFÞ
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e2
1 vT ðx; 0Þ ¼ xð4m1Þ=2 Tc vy
1 vT ðx; 0Þ ¼ xð2m1Þ=2 ðiiiÞ ðVHFÞ T Tc vy
(5)
(6)
c From Eq. (4) it is clear that the term Tðx;0ÞT is dimensionless. Tc m This means that the term x must be also dimensionless. The only case that the term xm is dimensionless is the case with m ¼ 0. Otherwise its units will be lengthm (the units of x are
ðx; 0Þ is length). From Eq. (5) it is found that the term 1 vT Tc dimensionless. This means that the term vyxð4m1Þ=2 must be also dimensionless. The units of x and y are length. Therefore the exponent m must be m ¼ 1=4. For the case (iii) it is found that the exponent should be m ¼ 1=2. In summary the boundary conditions 5a, 5b and 5c in Ref. [1] are valid only for the following conditions ðiÞ ðVWTÞ m ¼ 0
(7)
ðiiÞ ðVHFÞ m ¼ 1=4
(8)
ðiiiÞ ðVHFÞ m ¼ 1=2
(9)
However, the authors presented results for values of exponent m different from the above values. For example in Fig. 6 in Ref. [1] temperature profiles are presented for the case (i) for m ¼ 0:2, m ¼ 0 and m ¼ 0:2. The results for m ¼ 0:2 and m ¼ 0:2 are wrong. Taking into account all the above the results presented in Ref. [1] are doubtful.
reference
[1] Haq Rizwan Ul, Hamouch Zakia, Hussain ST, Mekkaoui Toufik. MHD mixed convection flow along a vertically heated sheet. Int J Hydrogen Energy 2017;42:15925e32.
Please cite this article in press as: Pantokratoras A, Comments on “MHD mixed convection flow along a vertically heated sheet” [Int J Hydrogen Energy 42 (2017) 15925e15932], International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.08.139